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Full text of "Mathematical And Physical Papers - Iii"

216             ON  THE CONDUCTION  OF HEAT IN  CRYSTALS.
isothermal surface passing through that point, that is, in a direction parallel to the radius vector drawn from the centre of the thermic sphere to the point of contact of a tangent plane drawn parallel to the isothermal surface at the point considered.
Now the tangency of two surfaces is evidently unchanged by derivation. Hence, in a crystal, if we have given the direc- , tion of the isothermal surface at any point, we may find that of the flow of heat by the following construction. In a direction parallel to the isothermal surface at the given point draw a tangent plane to the thermic ellipsoid, and join the centre with ;                    the point of contact: the flow of heat will take place in a direc-
i                    tion parallel to this joining line.    In other words, the flow of
|  ,                            heat will  take  place  in  a  direction  parallel  to  the  diameter
|                               which is conjugate to a plane parallel to the isothermal surface
I '                            at the given point.
j;;    I
| \        \                           14.    Conceive a plate bounded by parallel surfaces to be cut
}V        '                     from a crystal, and heat to be applied towards its centre; and
;'                              suppose the lateral boundaries sufficiently distant to produce no
!* '        j                     sensible influence on the result, so that we may regard the plate
jr                               as infinite.    In this case the auxiliary solid will likewise be an
i                               infinite  plate  bounded  by  parallel  surfaces.    Now  if heat  be
; !        ;                     supplied  according to  any law at  one  point  of such a plate,
I                               or at any number of points situated in the same normal, the
|                              isothermal  surfaces   will  be  surfaces   of revolution, having  the
1                              normal drawn through  the source or sources of heat for their
;                              axis, and the isothermal curves in which the parallel faces are
1 ,                           cut by the isothermal surfaces will be circles, having their centres in the points in which the faces are cut by the normal above-mentioned.
Hence, in a crystalline plate, if heat be supplied according to any law at one point, or at any number of points situated in a line parallel to the diameter of the thermic ellipsoid which is conju-
I                             gate to the planes of the faces, (a line which for brevity I will call
the line of sources,) any particular isothermal surface will be a surface generated by an ellipse which has its plane parallel to
;                             the faces, its centre in the line of sources, and its principal axes
parallel and proportional to those of the ellipse in which the thermic ellipsoid is cut by a plane parallel to the faces. In par-